Theorem rnun 4942

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 4939	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4735	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4741	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2158	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4545	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4545	. . 3 |- ran A = dom `' A
7		df-rn 4545	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3223	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2168	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1331   u. cun 3064   `'ccnv 4533   dom cdm 4534   ran crn 4535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by:  imaundi  4946  imaundir  4947  rnpropg  5013  fun  5290  foun  5379  fpr  5595  fprg  5596  sbthlemi6  6843  exmidfodomrlemim  7050